Generalized Fermat equation: A survey of solved cases

IF 0.8 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae Pub Date : 2025-04-03 DOI:10.1016/j.exmath.2025.125688

Ashleigh Ratcliffe, Bogdan Grechuk

{"title":"Generalized Fermat equation: A survey of solved cases","authors":"Ashleigh Ratcliffe, Bogdan Grechuk","doi":"10.1016/j.exmath.2025.125688","DOIUrl":null,"url":null,"abstract":"<div><div>Generalized Fermat equation (GFE) is the equation of the form <span><math><mrow><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>=</mo><mi>c</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></math></span>, where <span><math><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>r</mi></mrow></math></span> are positive integers. If <span><math><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>/</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>/</mo><mi>r</mi><mo><</mo><mn>1</mn></mrow></math></span>, GFE is known to have at most finitely many primitive integer solutions <span><math><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></math></span>. A large body of the literature is devoted to finding such solutions explicitly for various six-tuples <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></math></span>, as well as for infinite families of such six-tuples. This paper surveys the families of parameters for which GFE has been solved. Although the proofs are not discussed here, collecting these references in one place will make it easier for the readers to find the relevant proof techniques in the original papers. Also, this survey will help the readers to avoid duplicate work by solving the already solved cases.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125688"},"PeriodicalIF":0.8000,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Expositiones Mathematicae","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S072308692500043X","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Generalized Fermat equation (GFE) is the equation of the form

a x^{p} + b y^{q} = c z^{r}

, where

a, b, c, p, q, r

are positive integers. If

1 / p + 1 / q + 1 / r < 1

, GFE is known to have at most finitely many primitive integer solutions

(x, y, z)

. A large body of the literature is devoted to finding such solutions explicitly for various six-tuples

(a, b, c, p, q, r)

, as well as for infinite families of such six-tuples. This paper surveys the families of parameters for which GFE has been solved. Although the proofs are not discussed here, collecting these references in one place will make it easier for the readers to find the relevant proof techniques in the original papers. Also, this survey will help the readers to avoid duplicate work by solving the already solved cases.

查看原文本刊更多论文

广义费马方程：已解案例综述

广义费马方程（GFE）的形式为axp+byq=czr，其中a、b、c、p、q、r为正整数。如果1/p+1/q+1/r<；1，则已知GFE有至多有限个原始整数解（x,y,z）。大量的文献致力于为各种六元组（A,b,c,p,q,r）以及这些六元组的无限族找到这样的明确解。本文综述了求解GFE的参数族。虽然这里没有讨论证明，但将这些参考文献收集在一起将使读者更容易在原始论文中找到相关的证明技术。同时，这个调查将帮助读者通过解决已经解决的案件来避免重复工作。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Expositiones Mathematicae 数学-数学

CiteScore

1.30

自引率

0.00%

发文量

审稿时长

40 days

期刊介绍： Our aim is to publish papers of interest to a wide mathematical audience. Our main interest is in expository articles that make high-level research results more widely accessible. In general, material submitted should be at least at the graduate level.Main articles must be written in such a way that a graduate-level research student interested in the topic of the paper can read them profitably. When the topic is quite specialized, or the main focus is a narrow research result, the paper is probably not appropriate for this journal. Most original research articles are not suitable for this journal, unless they have particularly broad appeal.Mathematical notes can be more focused than main articles. These should not simply be short research articles, but should address a mathematical question with reasonably broad appeal. Elementary solutions of elementary problems are typically not appropriate. Neither are overly technical papers, which should best be submitted to a specialized research journal.Clarity of exposition, accuracy of details and the relevance and interest of the subject matter will be the decisive factors in our acceptance of an article for publication. Submitted papers are subject to a quick overview before entering into a more detailed review process. All published papers have been refereed.